cp's OEIS Frontend

A220419 Number of ways to write n=x+y (x>0, y>0) with 2x+1, 2y-1 and x^3+2y^3 all prime.

%I A220419 #18 Sep 22 2023 18:10:41
%S A220419 0,0,1,0,1,0,1,1,0,1,1,1,3,1,1,0,2,1,2,1,2,1,2,0,1,1,0,2,1,1,2,2,1,1,
%T A220419 2,1,1,4,2,1,1,1,3,2,1,0,1,0,1,1,1,1,1,0,3,3,1,4,1,1,0,4,2,2,3,0,1,3,
%U A220419 2,2,1,0,5,2,0,0,1,2,2,2,2,1,2,2,3,3,2,0,1,0,2,2,4,3,2,1,3,4,2,3
%N A220419 Number of ways to write n=x+y (x>0, y>0) with 2x+1, 2y-1 and x^3+2y^3 all prime.
%C A220419 Conjecture: a(n)>0 for all n>527.
%C A220419 This has been verified for n up to 2*10^7. It implies the Goldbach conjecture since 2(x+y)=(2x+1)+(2y-1).
%C A220419 Zhi-Wei Sun also made the following similar conjectures:
%C A220419 (1) Each integer n>1544 can be written as x+y (x>0, y>0) with 2x-1, 2y+1 and x^3+2y^3 all prime.
%C A220419 (2) Any odd number n>2060 can be written as 2p+q with p, q and p^3+2((q-1)/2)^3 all prime.
%C A220419 (3) Every integer n>25537 can be written as p+q (q>0) with p, p-6, p+6 and p^3+2q^3 all prime.
%C A220419 (4) Any even number n>1194 can be written as x+y (x>0, y>0) with x^3+2y^3 and 2x^3+y^3 both prime.
%C A220419 (5) Each integer n>3662 can be written as x+y (x>0, y>0) with 3(xy)^3-1 and 3(xy)^3+1 both prime.
%C A220419 (6) Any integer n>22 can be written as x+y (x>0, y>0) with (xy)^4+1 prime. Also, any integer n>7425 can be written as x+y (x>0, y>0) with 2(xy)^4-1 and 2(xy)^4+1 both prime.
%C A220419 (7) Every odd integer n>1 can be written as x+y (x>0, y>0) with x^4+y^2 prime. Moreover, any odd number n>15050 can be written as p+2q with p, q and p^4+(2q)^2 all prime.
%C A220419 Conjectures (1) to (7) verified up to 10^6. - _Mauro Fiorentini_, Sep 22 2023
%H A220419 Zhi-Wei Sun, <a href="/A220419/b220419.txt">Table of n, a(n) for n = 1..20000</a>
%H A220419 D. R. Heath-Brown, <a href="https://doi.org/10.1007/BF02392715">Primes represented by x^3 + 2y^3</a>. Acta Mathematica 186 (2001), pp. 1-84.
%H A220419 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588 [math.NT], 2012-2017.
%e A220419 a(25)=1 since 25=3+22 with 2*3+1, 2*22-1 and 3^3+2*22^3=21323 all prime.
%e A220419 a(26)=1 since 26=11+15 with 2*11+1, 2*15-1 and 11^3+2*15^3=8081 all prime.
%t A220419 a[n_]:=a[n]=Sum[If[PrimeQ[2k+1]==True&&PrimeQ[2(n-k)-1]==True&&PrimeQ[k^3+2(n-k)^3]==True,1,0],{k,1,n-1}]
%t A220419 Do[Print[n," ",a[n]],{n,1,1000}]
%Y A220419 Cf. A220413, A173587, A220272, A219842, A219864, A219923.
%K A220419 nonn
%O A220419 1,13
%A A220419 _Zhi-Wei Sun_, Dec 14 2012